Exponential stability and fixed-time control of a stochastic avian influenza model with spatial diffusion and nonlocal delay

Spatial heterogeneity, random disturbances in the external environment, and the incubation period of infected individuals collectively have a significant impact on the outbreak of avian influenza. In this paper, a stochastic susceptible-infective-susceptible-infected-recovered (SI-SIR) avian influenza model is established that incorporates spatial diffusion and nonlocal delay. The existence and uniqueness of mild solutions are established by applying the Banach fixed point theorem, the truncation method, and the semigroup approach. Based on the Borel-Cantelli lemma, the mean-square exponential stability and almost sure exponential stability of the mild solution are analyzed. Additionally, in combination with the Lyapunov theory, a fixed-time control strategy is proposed to achieve stability within the desired settling time. Numerical simulations are conducted to validate the impacts of key parameters and enhance the understanding of the results of the theory.